On group codes arising from Paley-type partial difference sets and skew–Hadamard difference sets

Abstract Paley-type partial difference sets and skew–Hadamard difference sets are classical objects in algebraic combinatorics, known for their rich connections with graph theory, coding theory, and group theory. In this paper, we explore new links between these combinatorial structures and group codes arising as ideals in finite group algebras. We construct such codes from difference sets and determine their dimensions in several cases. As an application of our links, we explicitly compute the full set of primitive central idempotents in certain abelian $$ p $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -group algebras, by employing the classical sets of quadratic residues and non-residues modulo $$ p $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> , which are well-studied examples of difference and partial difference sets—we also obtain their dimensions and estimate their minimum weights.